Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, February 2011."May 2010." Cataloged from PDF version of thesis.Bow-springs find few applications in industry. Principally, they are used in archery. In addition, they have found some use in a compression-spring mode in the field of biomechatronics, to emulate elastic human legs. The mechanical behavior (characterized by deflected shape and deformation force) is difficult to model, because internal forces and moments and the geometry are both unknown. The only closed-form solutions to such systems are relatively useless to a mechanical engineer. This work comprises an iterative model developed in MATLAB that computes the mechanical behavior of buckled beam (or bow-spring) sections, over a range of parameters and geometries, to be used in the development and testing of compression bow-springs as parallel loading systems to the human leg.by Blake A. Sessions.S.B

Sessions, Blake A

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Blake SessionsA Computational Bow-spring ModelMASSACby OFBlake A. Sessions JB.S. Mechanical Engineering LIMassachusetts Institute of Technology, 2011HUSETTS INSTITUTETECHNOLOGYN 3 0 2010BRARIESSUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING IN PARTIALFULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OFBACHELOR OF SCIENCE IN MECHANICAL ENGINEERINGAT THEMASSACHUSETTS INSTITUTE OF TECHNOLOGY ARCHIVESMAY 2010@2010 Blake A. Sessions. All rigFts reserved.The author hereby grants to MIT permission to reproduce and to distribute publiclypaper and electronic copies of this thesis document in whole or in part in any mediumnow known or hereafter created.Signature of Author:Certified by:k -~ *~'Accepted by:Departrff4of Mecha~lical EngineeringMay 3, 2010Hugh HerrProfessor, Media Arts and SciencesThesis SupervisorCollins Professor of MecJohn H. Lienhard V:hanical EngineeringChairman, Undergraduate Thesis Committeep. 1Blake SessionsA Computational Bow Spring ModelbyBlake A. SessionsSubmitted to the Department of Mechanical Engineeringon May 7, 2010 in Partial Fulfillment of theRequirements for the Degree of Bachelor of Science inMechanical EngineeringABSTRACTBow-springs find few applications in industry. Principally, they are used in archery. Inaddition, they have found some use in a compression-spring mode in the field ofbiomechatronics, to emulate elastic human legs. The mechanical behavior(characterized by deflected shape and deformation force) is difficult to model, becauseinternal forces and moments and the geometry are both unknown. The only closed-form solutions to such systems are relatively useless to a mechanical engineer.This work comprises an iterative model developed in MATLAB that computes themechanical behavior of buckled beam (or bow-spring) sections, over a range ofparameters and geometries, to be used in the development and testing of compressionbow-springs as parallel loading systems to the human leg.Thesis Supervisor: Hugh HerrTitle: Professor, Media Arts andSciences0.4 -03The set of spatial curves that a \thin bow-spring will assume intwo-force-membercompression, iterated to 60%deformationp. 2Blake SessionsIntroductionBowed members are difficult to analyze mechanically, because the compatibilityarguments (and overall geometry) are intricately convolved with the internal forces andmoments. Standard mechanical analyses (for example - a beam undergoing a torsionalmoment, or a cantilevered and loaded beam) of internal forces and moments aretypically very straight-forward, and require no more technical aptitude than second-ordersingle variable calculus.Bowed springs, however, are a different story: The geometry is such that itcannot be simplified into a single differential equation that can be integrated. There isvery little established in current literature that correctly defines the shape and force-deformation behavior of a bowed spring (given the geometry, elastic modulus, and othermaterial properties.)Using standard mechanical analysis techniques, we could start with acompatibility argument. Subjecting our bow of length L to an end-to-end deformation Dwould cause the bow to curve in order to fit compatibly. Unfortunately, there are aninfinite number of spatial curves that the bow could assume to satisfy this compatibility.Only one satisfies static compatibility, but this does not give us enough information tosolve the problem. The shape is not parabolic in nature, nor is it an arc segment. Thus,we do not know how far the center point of the bow lies from the loading axis (if weknew this, we could then simply find the internal moment and divide by the distance tothe loading axis to find the force output.)This is a classic problem of Calculus of Variations: We are picking the particularspatial curve r(s) that results in the minimum elastic energy storage in the beam. Anyp. 3Blake Sessionsdynamics or oscillations will be about this equilibrium point. The analytic solution,however, results in an elliptical integral: the only closed form solution to the deformedshape of bow-spring sections. However, the degrees of freedom used to express thestate of the elastically deformed section are such that they may not be translated intoCartesian geometry. Although the closed form may be of help intellectually, it is of verylittle use practically.This work comprises the development of a numerical system that describes thebow-spring behavior, which can be articulated in two manners. The end-to-end force asit relates to the deformation is incredibly useful to the engineer, because from it we canmodel the spring as a simple variable impedance spring (as a function of deformation orload.)In addition, the physical shape that the bow-spring assumes under load is equallyuseful, because from it we can find the maximum internal moment and curvature in thematerial (which resides at the center of the spring, if statically loaded.) From theseparameters and our simple linear stress-strain techniques, we may find the maximuminternal stress in the material and define safety factors to ensure that the material willnot fail in application.p. 4Blake SessionsStructure of the solutionThe initial assumption, from which the rest of the derivation is based, resides inthe fact that the internal moment of the bow will be proportional to its displacement fromthe two-force axis. This is immediately apparent if an internal slice of the beam is taken;at any section, there is an internal moment M which is equal to F*y, where F is the end-to-end spring force, and y is the perpendicular distance to the load axis.Because the spring force does not change with the location of our cut, I find thatMint(y) = F*y, a direct proportionality. The internal moment and the curvature are linearlyrelated, viz.Mint = EI*KSo the internal curvature is also proportional to the distance from the load axis,K(y) = (F/EI)*yWhen making the compatibility argument, we exploit this proportionality. Because wedon't know the end-to-end force F, we settle for an unknown constant:K(y) = B*yThus, we find that the curvature at each particular point along the length of the sectionis proportional to its distance from the load axis.Iteration can then be accomplished, by subjecting the beam to a curvature whichis proportional to y. We must know the initial angle that the beam makes with the loadaxis (at the load point) in order to iterate from the end of the beam. Because we don'tyet know this, we must choose to iterate from the center. By symmetry, we know theinitial angle that the beam makes with the load axis: It is the angle defined in the rigidknee, divided by two.p. 5Blake SessionsHowever, we have yet another problem. Starting from the center of the beam,the code must iterate outwards with a curvature proportional to the distance to the loadaxis. At which y value does the load axis reside? We don't yet know, so we defineanother arbitrary constant Ymo, the distance from the center of the beam to the loadaxis.If guesses of "B" and "Ymo" are made, we can find a particular bow spring curve;one that could possibly exist, somewhere. The geometry varies as a function of B andYmo, which are the initial parameters for the iteration. Lowercase x,y represent the pointlocations of the bow in cartesian space, with the origin set at the midpoint of the bow.Lowercase "I" is the length of the bow from the center-point as the code iterates,otherwise often called "s" for path length. Theta is the angle that is made with thehorizontal axis. The capital versions of these variables are storing the instantaneousvariables as vectors for later use. The interior-most loop continues to iterate until thecurve reaches Ymo, the loading axis.There exists a major problem with this iteration, however. Although it hasproduced a bow-spring curve, it is not the curve that is desired. The length and thedeformation of the bow-spring vary as a function of B, Ymo. I prefer to use non-dimensional deformation, defined as e = d/L; the spring deformation divided by thelength of the spring. Then:(e, L) = F(B, Ymo)Where each variable e,L has a dependence upon the other two variables B, Ymo. Ahighly non-linear system, the dependent variables e,L cannot be put into a linear matrixform. Thus, we resort to another method: we assume that the non-dimensional shapep. 6Blake Sessionsof bow-springs remains constant. If we have two bow springs, one of length L with adeformation d and one of length 2*L and a deformation 2*d, the spatial curve that thelatter assumes will be a linear scale factor of 2 of the initial geometry. This can be seenfrom an energetics perspective; if we proceed to attack the problem with a calculus ofvariations approach (as stated in the introduction), dimensionless parameters pop out ofthe system that are linear scales of the contained elastic energy, and are not dependentupon the non-dimensional shape.We then define "ed", the desired non-dimensional deformation. The code runsthrough an array of Ymo until the curve satisfies the non-dimensional shape requirement.Each time, it resets the state variables x, y, I, and theta. Once the outer iteration loophits the correct ed value, the bow-spring curve has been non-dimensionally defined. Itthen take the vectors X and Y (which have been storing the instantaneous values), andscale them accordingly so that the curve assumes its proper size.The next lines of code are geometric arguments to find the internal curvature ofthe beam, at the center point (for every deformation and length, one value of maximumcurvature is defined.) Internal moments and end-to-end forces are easily derived fromthis set of data. The outermost loop runs through a deformation vector and stores theinformation from each loop iteration. Finally, some offsetting code is implemented at theend to make the F-d relation more apparent and to cut off the "tail."As further proof of the non-dimensional assumption and scaling factor argument,we find that the force-deformation curve is not dependent upon the initial scaling factor,B. B can be defined from 0.1 to 5 with little effect; small errors of order 1% are apparentdue to the larger steps of iteration. Thus, the initially defined parameters B, Ymo do notp. 7Blake Sessionsaffect the output curve, which is what we expect. The "inputs" have been iterated awayto become intermediate variables to the force output.Using the codeInputs are:El bending stiffness (N*m2)Ltot end-to-end length of bow-spring (m)dmax the maximum deformation (m)KAngle the knee angle, or intersection angle of unstressed springs (radians)n number of iterations for the force-deformation curve.Outputs are:def end-to-end deformation (m) (vector)Fout end-to-end force (N) (vector)(d,F) plotted by default.p. 8Blake SessionsRemarksIntuitively, we expect a few things from bow-spring behavior. We expect that theoutput force will be linearly related to the bending stiffness. For a defined deformation"d", we expect the output force to decrease as we choose larger and larger springsections, because the non-dimensional shape gets progressively "less bowed" as wechoose larger springs. We expect the curve to be independent of the number ofdeformation iterations. All of the above can be seen in the model.The effect of the knee angle upon the force-deformation relation is moreinteresting. With a shallower (closer to parallel) knee angle, we expect a spring that isinitially stiffer. Furthermore, we expect that the behavior of a set bow-springs with thelength held constant and the knee angle varied to converge. At low deformations, a 10-degree knee angle will result in a very stiff leg, because our initial force input will beresisted at a small radius, which yields a small internal moment and a small curvature.A 40-degree knee angle will result in a substantially increased compliance, for a similarreason. However, at large deformations, the compliances of the 10-degree and 40-degree systems should converge; Both springs have been displaced substantially fromthe neutral axis, and the "memory" of the system has lowered at large deformations. Inother words, the compliance has a large functional dependence upon knee angle at lowdeformations, and a negligible dependence upon knee angle at high deformations.Below are three force-deformation plots, with initial parametersEl = 120, L = 1, dmax = 0.15, and three different knee angles KAngle = 0.02,0.05,0.1radians. Notice that the compliances (values represented by the initial slope) arep. 9Blake Sessionsdrastically different at small deformation, and converge to the same value at largedeformation. (units: X - meters, Y - Newtons.)ComparisonsAlena and Beth tested the force-deformation behavior of various sizes of bow-spring. Using the model, the inputs required for a force-deformation curve are bendingstiffness, length, deformation range, and knee angle. With the proper parametersimplemented, the following curves result for the bow-springs used in the hopping testsof the following people: Alena, Caitlin, Hazel, Andy, Ian, and Karin. Data represents avariety of spring thicknesses, from 0.25" to 0.31", and a range of spring lengths from 33"to 37." All plots have units X - meters, Y - Newtons. Green: model, Blue: measured.p. 10Blake Sessionsp. 11Blake Sessionsp. 12Blake Sessionsp. 13Blake SessionsError AnalysisThere exist three principal sources of error. The first is a non-linearity in thebending stiffness of the beams; for all of these loadings, we assume an elastic loadingwith a stress proportional to the distance to the neutral axis. In reality, this isn't quite thecase, and the force-deformation curves don't scale linearly with the bending stiffness,El. Generally, the linearity dies away as we get farther from the neutral axis of the bentbeam; thus, the internal stress distribution will not produce as great an internal momentas the model predicts for any particular defined curvature. This will result in a springwhich, in reality, will exhibit a lower force output. These errors are particularly apparentin the cases with thicker springs (Caitlin, Ian, Andy.)Secondly, the model assumes elasticity up to the center-point of the bow. Everypart of the bow produces a curvature when loaded. In reality, we have a rigid kneelocking the two carbon fiber sections together, which has a mechanical impedance thatis much greater than the springs themselves. When we load the beam, this rigid sectionwill not deform as the model predicts, and will have a greater stiffness in reality than inthe predicted model. Particularly for the clutched system (whose knee is large,) the legwill exhibit a much higher F-d curve in reality.Finally, error can be attributed to the asymmetry of the leg. The model predicts aperfect symmetry of the bowed section, and in reality the knee is off-center. We mayderive some insight into this effect by looking at the end condition, as the knee nearsone of the pivots. In this case, the material distribution (with the knee angle heldconstant) will be much closer to the loading axis, and it will be harder to deformoutwards. Thus, the spring in reality will exhibit a higher impedance when the knee isoff-center. With these considerations in mind, the model still yields a result that isgenerally accurate to -10% or less, and may prove to be useful in the future.p. 14Blake SessionsAppendix A - MATLAB code%BOW SPRING MODEL. UNITS IN SI UNLESS NOTED.EI = 66; %Bending stiffness. N*m^2Ltot = 1; %BS length, end to end.dmax = 0.3;KAngle = 0.25; %From horizontal to one side length.n = 10; %number of linear deformation iterationsthetao = KAngle/2;i = 1;Ld = Ltot;def=O;Fout=0;B = 0.5; %Arbitrary initial constant.for ed = 0.0001/Ld:dmax/(n*Ld):dmax/Ld; %desired non-dimensionaldeformation.e = 0; %for the first iterationYmo 0;X = 0;Y = 0;L = 0;Theta = 0;while e <= edYmo = Ymo + 0.01;x = 0;y = 0;1 = 0;theta = thetao;dL = 0.001;j = 1;while y <= Ymo;%DIFFERENTIALSdtheta = B*(Ymo-y)*dL;dx = dL*cos(theta);dy = dL*sin(theta);%UPDATING STATE VARIABLEStheta = theta + dtheta;p.15Blake Sessionsx = x + dx;y = y + dy;1 = 1 + dL;X(j) = x; %FOR plotting.Y(j) = y;Theta(j) = theta;L(j) = 1;j = j+1; %FOR NEXT ITERATION w/ unknown lengthende = 1-x/l;end%SCALINGX = X*Ld/(2*l);Y = Y*Ld/(2*l);L = L*Ld/(2*l);%CURVE HAS BEEN DEFINED COMPLETELY AS A FUNCTION OF DEFORMATION.%Finding the internal curvature. using points 2,3,4.dxl=X(1,26)-X(1,2);dx2=X(1,50)-X(1,26);dy1=Y(1,26)-Y(1,2);dy2=Y(1,50)-Y(1,26);phil = atan(dyl/dxl);phi2 = atan(dy2/dx2);dphi=phi2-phil;difflel = sqrt(dxlA2+dyl^2);diffle2 = sqrt(dx2A2+dy2^2);diff=(difflel + diffle2);curvature = dphi/diff;%plotting the curve of the bowspring, scaled properly.plot(X,Y,'b')hold onplot(-X,Y,'b')hold onp. 16Blake Sessionsaxis equal%for calculating internal moment.Mint(i)=EI*curvature;Yend(i)=Y(end);i = i+1;enddeformation = 0.0001:dmax/n:dmax; %Deformation output vector.F = Mint./Yend; %Force output vector.%Compensating for the tail. find the number of pts before def.force, then%define new vectors.j = 1;while F(j) <= 1 %Discard force.Fout(1) = F(j);j = j+1;end%Reassignment.for k = 1:1:(n-j)Fout(k)=F(k+j);def(k) = deformation(k+j);internalmoment(k) = Mint(k+j);end%Shifting curve to zero point.offset=def(l);def = def - offset;%F-d plot.%plot(def,Fout,'r*')%hold onp.17

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